CHAPTER 4 Structure of the Atom 4.1 The Atomic Models of Thomson and Rutherford 4. Rutherford Scattering 4.3 The Classic Atomic Model 4.4 The Bohr Model of the Hydrogen Atom 4.5 Successes & Failures of the Bohr Model 4.6 Characteristic X-Ray Spectra and Atomic Number 4.7 Atomic Excitation by Electrons The opposite of a correct statent is a false statent. But the opposite of a profound truth may well be another profound truth. An expert is a person who has made all the mistakes that can be made in a very narrow field. Never express yourself more clearly than you are able to think. Prediction is very difficult, especially about the future. Niels Bohr (1885-196) - Niels Bohr Structure of the Atom Evidence in 19 indicated that the atom was not a fundantal unit: 1) There seed to be too many kinds of atoms, each belonging to a distinct chemical elent (way more than earth, air, water, and fire!). ) Atoms and electromagnetic phenona were intimately related (magnetic materials; insulators vs. conductors; different emission spectra). 3) Elents combine with so elents but not with others, a characteristic that hinted at an internal atomic structure (valence). 4) The discoveries of radioactivity, x rays, and the electron (all seed to involve atoms breaking apart in so way). Knowledge of atoms in 19 Thomson s Atomic Model Electrons (discovered in 1897) carried the negative charge. Electrons were very light, even compared to the atom. Protons had not yet been discovered, but clearly positive charge had to be present to achieve charge neutrality. Thomson s plum-pudding model of the atom had the positive charges spread uniformly throughout a sphere the size of the atom, with electrons embedded in the uniform background. In Thomson s view, when the atom was heated, the electrons could vibrate about their equilibrium positions, thus producing electromagnetic radiation. Unfortunately, Thomson couldn t explain spectra with this model. Experints of Geiger and Marsden Rutherford, Geiger, and Marsden conceived a new technique for investigating the structure of matter by scattering particles from atoms. Experints of Geiger and Marsden Geiger showed that many particles were scattered from thin gold-leaf targets at backward angles greater than 9. 1
Electrons can t backscatter particles. Before After Calculate the maximum scattering angle corresponding to the maximum montum change. Try multiple scattering from electrons If an particle is scattered by N electrons: N the number of atoms across the thin gold layer, t 6 1 7 m: n It can be shown that the maximum montum transfer to the particle is: p m max ev Determine θ max by letting p max be perpendicular to the direction of motion: The distance between atoms, d n -1/3, is: θ max p p v M v too small! N t / d still too small! Rutherford s Atomic Model 4.: Rutherford Scattering even if the particle is scattered from all 79 electrons in each atom of gold. Experintal results were not consistent with Thomson s atomic model. Rutherford proposed that an atom has a positively charged core (nucleus) surrounded by the negative electrons. Geiger and Marsden confird the idea in 1913. Ernest Rutherford (1871-1937) Scattering experints help us study matter too small to be observed directly. There s a relationship between the impact parater b and the scattering angle θ. When b is small, r is small. the Coulomb force is large. θ can be large and the particle can be repelled backward. where K 1 mv cot(θ/) π θ The cross section σ π b is related to the probability for a particle being scattered by a nucleus (t foil thickness): The fraction of incident particles scattered is: The number of scattering nuclei per unit area: Rutherford Scattering Any particle inside the circle of area π b will be similarly (or more) scattered. Rutherford Scattering Equation The number of particles scattered per unit area is: In actual experints, a detector is positioned from θ to θ + dθ that corresponds to incident particles between b and b + db.
Rutherford scattering experint 1 MeV protons scattering off gold foil. Note the correct dependence on scattering angle. 4.3: The Classical Atomic Model Consider an atom as a planetary system. The Newton s nd Law force of attraction on the electron by the nucleus is: 1 e mv Fe r r where v is the tangential velocity of the electron: e v e 1 1 K mv mr r The total energy is then: This is negative, so the system is bound, which is good. The Planetary Model is Dood 4.4: The Bohr Model of the Hydrogen Atom From classical E&M theory, an accelerated electric charge radiates energy (electromagnetic radiation), which ans the total energy must decrease. So the radius r must decrease!! Electron crashes into the nucleus!? Bohr s general assumptions: 1. Stationary states, in which orbiting electrons do not radiate energy, exist in atoms and have well-defined energies, E n. Transitions can occur between them, yielding light of energy: E E n E n hν. Classical laws of physics do not apply to transitions between stationary states, but they do apply elsewhere. n n 1 n 3 Physics had reached a turning point in 19 with Planck s hypothesis of the quantum behavior of radiation, so a radical solution would be considered possible. 3. The angular montum of the n th state is: nħ where n is called the Principal Quantum Number. Angular montum is quantized! Consequences of the Bohr Model Bohr Radius The angular montum is: L mvr nħ So the velocity is: v n ħ / mr But: v e mr So: nħ m r e mr a The Bohr radius, ħ a is the radius of the unexcited hydrogen atom and is equal to: Solving for r n : rn n a where: ħ a The ground state Hydrogen atom diater is: a is called the Bohr radius. It s the diater of the Hydrogen atom (in its lowest-energy, or ground, state). 3
The Hydrogen Atom Energies Use the classical result for the e E energy: 8πε r and: r nħ n So the energies of the stationary states are: The Hydrogen Atom Emission of light occurs when the atom is in an excited state and decays to a lower energy state (n u n l ). 1 ν hν λ c hc hν Eu E l where ν is the frequency of a photon. or: E n E /n where E 13.6 ev. R is the Rydberg constant. R 4 3 (4 πħ) cε Transitions in the Hydrogen Atom The atom will remain in the excited state for a short ti before emitting a photon and returning to a lower stationary state. In equilibrium, all hydrogen atoms exist in n 1. 4.6: Characteristic X-Ray Spectra and Atomic Number Shells have letter nas: K shell for n 1 L shell for n The atom is most stable in its ground state. An electron from higher shells will fill the inner-shell vacancy at lower energy. When it occurs in a heavy atom, the radiation emitted is an x-ray. It has the energy E (x ray) E u E l. Atomic Number and Moseley The x-rays have nas: L shell to K shell: K x-ray M shell to K shell: K β x-ray etc. K β K Moseley s Empirical Results The K x-ray is produced from the n to n 1 transition. In general, the K series of x-ray wavelengths are: G.J. Moseley studied x-ray emission in 1913. Atomic number Z number of protons in the nucleus. Moseley found a relationship between the frequencies of the characteristic x-ray and Z. Moseley found this relation holds for the K x-ray: ν K 3cR ( Z 1) 4 We use Z-1 instead of Z because one electron is already present in the K-shell and so shields the other(s) from the nucleus charge. Moseley s research clarified the importance of Z and the electron shells for all the elents, not just for hydrogen. 4
The Correspondence Principle Bohr s correspondence principle is rather obvious: The Correspondence Principle The frequency of the radiation emitted ν classical is equal to the orbital frequency ν orb of the electron around the nucleus. ν classical ω ν v / r ν orb classical π π This should agree with the frequency of the transition from n + 1 to n (when n is very large): ν Bohr In the limits where classical and quantum theories should agree, the quantum theory must reduce the classical result. For large n: ν Bohr Substituting for E : ν Bohr ν classical 4.7: Atomic Excitation by Electrons Franck and Hertz studied the phenonon of ionization. Atomic Excitation by Electrons Ground state has E to be zero. First excited state has E 1. The energy difference E 1 E 1 is the excitation energy. Hg has an excitation energy of 4.88 ev in the first excited state No energy can be transferred to Hg below 4.88 ev because not enough energy is available to excite an electron to the next energy level Accelerating voltage is below 5 V: electrons did not lose energy. Accelerating voltage is above 5 V: sudden drop in the current. Above 4.88 ev, the current drops because scattered electrons no longer reach the collector until the accelerating voltage reaches 9.8 ev and so on. Fine Structure Constant The electron s velocity in the Bohr model: v n In the ground state, v 1. 1 6 m/s ~ 1% of the speed of light. 4.5: Successes and Failures of the Bohr Model Success: The electron and hydrogen nucleus actually revolve about their mutual center of mass. The electron mass is replaced by its reduced mass: The ratio of v 1 to c is the fine structure constant. v 1 c The Rydberg constant for infinite nuclear mass, R, is replaced by R. 5
Limitations of the Bohr Model The Bohr model was a great step in the new quantum theory, but it had its limitations. Failures: Works only for single-electron ( hydrogenic ) atoms. Could not account for the intensities or the fine structure of the spectral lines (for example, in magnetic fields). Could not explain the binding of atoms into molecules. 6